propositional logic

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propositional logic

A student writes a simple statement in propositional logic on a whiteboard.

Definition
  1. Noun:
    • A formal system of logic: Propositional logic is a branch of symbolic logic. Its fundamental units are propositions, which are declarative statements that can be either true or false. It studies how these propositions can be combined using logical connectives (like "and," "or," "not," "if...then") to form more complex statements, and how the truth or falsity of the complex statement depends on the truth values of its component parts.
Usage
  • As a subject of study: Propositional logic is foundational in mathematics, computer science, and philosophy.
    • Understanding propositional logic is essential for designing digital circuits.
    • The course begins with the basics of propositional logic before moving to predicate logic.
  • As a tool or method: It is used as a formal system for analyzing arguments and reasoning.
    • We can translate this argument into propositional logic to check its validity.
    • The proof was verified using the rules of propositional logic.
Advanced Usage
  • "Classical propositional logic": Refers to the standard, two-valued system where every proposition is either true or false.
    • The law of excluded middle is a fundamental axiom of classical propositional logic.
  • "Propositional logic calculus": Refers to the formal system with defined symbols, formation rules, and inference rules.
    • The completeness theorem for the propositional logic calculus was proven by Emil Post.
Variants and Related Words
  • Propositional calculus (n): A synonymous term for propositional logic, emphasizing its formal, calculative nature.
    • The syntax of the propositional calculus is defined recursively.
  • Sentential logic (n): Another synonymous term, where "sentence" is used interchangeably with "proposition."
    • Sentential logic deals with the logical relationships between entire sentences.
  • Predicate logic (n): A more expressive system of logic that extends propositional logic by analyzing the internal structure of propositions, including quantifiers ("for all," "there exists").
    • While propositional logic treats "Socrates is mortal" as a single symbol, predicate logic can break it down into a subject and a property.
Synonyms
  • Sentential calculus
  • Statement logic
  • Zero-order logic
Related Concepts (Not Phrasal Verbs/Idioms)
  • Logical connective: An operator (e.g., ∧ for "and," ∨ for "or," ¬ for "not," → for "implies") used to combine propositions.
    • The truth table defines the meaning of each logical connective.
  • Truth table: A tabular method used in propositional logic to determine the truth value of a complex statement for all possible combinations of truth values of its components.
    • We used a truth table to prove the tautology.
  • Tautology: A statement in propositional logic that is true under every possible interpretation (every row of its truth table is true).
    • "P or not P" is a classic example of a tautology.
  • Contradiction: A statement that is false under every possible interpretation.
    • "P and not P" is a fundamental contradiction.
propositional logic

A student writes a simple statement in propositional logic on a whiteboard.

Noun
  1. a branch of symbolic logic dealing with propositions as units and with their combinations and the connectives that relate them

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